Regularity of Fourier integral operators with amplitudes in general Hörmander classes

Abstract

We prove the global $$L^p$$ L p -boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes $$S^{m}_{\rho , \delta }(\mathbb {R}^n)$$ S ρ , δ m ( R n ) for parameters $$0\le \rho \le 1$$ 0 ≤ ρ ≤ 1 , $$0\le \delta <1$$ 0 ≤ δ < 1 . We also consider the regularity of operators with amplitudes in the exotic class $$S^{m}_{0, \delta }(\mathbb {R}^n)$$ S 0 , δ m ( R n ) , $$0\le \delta < 1$$ 0 ≤ δ < 1 and the forbidden class $$S^{m}_{\rho , 1}(\mathbb {R}^n)$$ S ρ , 1 m ( R n ) , $$0\le \rho \le 1.$$ 0 ≤ ρ ≤ 1 . Furthermore we show that despite the failure of the $$L^2$$ L 2 -boundedness of operators with amplitudes in the forbidden class $$S^{0}_{1, 1}(\mathbb {R}^n)$$ S 1 , 1 0 ( R n ) , the operators in question are bounded on Sobolev spaces $$H^s(\mathbb {R}^n)$$ H s ( R n ) with $$s>0.$$ s > 0 . This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.

An $ L^{q}\rightarrow L^{r} $ estimate for rough Fourier integral operators and its applications

“…The key breakthrough of their work was to show that the FIOs with a ∈ S −(n−1)/2 1,0 are locally bounded from the Hardy spaces H 1 (R n ) to Lebesgue spaces L 1 (R n ). For the symbols in general Hörmander class we refer to [2,12,20,21] . As for the problem of global boundedness of FIOs, we want to mention the operators with the symbol a in different classes, such as the SG classes by Cordero, Nicola and Rodino [5], the certain subclass of S 0 1,0 by Coriasco and Ruzhansky [6] and the FL p spaces by Cordero, Nicola and Rodino [4].…”